Cost, revenue and profit functions
Please Complete the questions attached. Each question must showing the working out and must follow every instruction.Applications of functions in finance, e.g. cost, revenue and profit functions.Supply and demand curves; market equilibrium and elasticity.Simple and compound interest; interest rates.Applications of geometric series, e.g. annuities; sinking funds; mortgage and debt repayments.Applications of differentiation, e.g. marginal functions; price discrimination models; Lagrangian multiplier and shadow price.Applications of integration, e.g. consumers’ and producers’ surplus.Linear inequalities and the graphical solution to the two variable linear programming problem; basic concept of the simplex algorithm.
-Input Output analysis.
-Definitions and background of quality assurance.
-Statistical principles of sampling and estimation.
-Introduction to QMS’s (Quality Management Systems).
-Ishikawa’s seven Quality tools.
-Process Capability Analysis.
-Control charts for measured variables: Shewhart (Xbar-R) Charts, Time weighted control charts (EWMA and MA charts).
Q.1 On her 25th birthday a woman opens a retirement saving account that pays a 3.2% annual interest rate. She has decided to pay /SOO into the account at the beginning of each quarter.
(a) How much will she have in her retirement pot if she decides to retire at the age of 60?
(b) Once retired she would like an annual income of £19700. Calculate how many years her retirement pot will last, assuming that the annual interest rate will be 5% for the period after retirement.
(c) On reaching the age of 60 she changes her mind about retirement and finds a different pension scheme that pays a 5% annual interest rate on sums over 150000. She plans to continue with £800 quarterly payments. If she retires at the age of 65, what would her annual income be for the next 20 years? Assume equal annual payments and that the annual interest rate will stay at 5%.
Q.2 An investor has decided to extend their investment portfolio by spending no more than £5000 on shares in companies A and B. The cost of shares is £5 and £10, respectively. The investor's financial advisor has estimated that they can expect to make a profit of La on each share in company A and Lb on each share in company B. Due to environmental concerns the investor has decided to buy at least twice as many shares in company B as in company A.
(a) Formulate the linear programming problem for maximising the profit.
(b) Sketch the feasible region.
(c) How many of each type of the share should the investor buy to maximise the profit if a = £0.80 and b = £0.65 ? State the corresponding profit.
(d) What condition do a and be 6 need to satisfy so that the optimal number of shares stays the same, as in solution for (c)?
Q.3 (a) The supply and demand functions of a product are given by P = 32 + Q2 P = 140 — Q2/3 where P denotes the price and Q the quantity.
(i) Sketch both of these functions on the same graph.
ii) Calculate the producer's and consumer's surpluses at the equilibrium point.
(iii) If the government imposes a fixed tax on this product, explain the effect on: A. equilibrium point: B. consumer's surplus.
(b) The price elasticity of demand is given by E = 2—QI, where P denotes the price and Q the quantity demanded. It is known that quantity demanded is 5 when the price is 10. Find the expression for the demand function.
A firm produces two different kinds of commodities, A and B. By selling Q1 tons of the first commodity the firm gets a price per ton given by P = 84 — 2Q1. By selling Q2 tons of the other commodity the price per ton is given by P = 96 — 402. The total cost of producing Q1 tons of A and 02 tons of B is C(Q1. 02) = + 20102 + 20?.
(a) Write down the revenue function and the profit function.
(b) Find the production levels that maximise the profit.
(c) Suppose that the firm's production activity causes so much pollution so that the authorities limit the output to 11 tons in total. Solve the firm's maximisation problem in this case. What is the loss in profit?